Sudoku Grids and Related Combinatorial Structures

This is an ongoing collaboration between Sian K. Jones, Stephanie Perkins, Paul A Roach, and Daniel Williams. In 2006 at the height of the interest in Sudoku puzzles this research area formed motivated by Sian K. Jones’ PhD in Sudoku grids.
A number of areas are explored, including the structure and properties of these grids, the number of these grids and their related applications.

Related publications:

S. K. Jones and S. Perkins. A P-box and S-box approach to encryption using Sudoku grids. Submitted 2016.

S. K. Jones, P. A. Roach, and S. Perkins. On the number of 6 × 6 sudoku grids. Journal of Combinatorial Mathematics and Combinatorial Computing, 89:33-44, August 2014.

S. K. Jones, S. Perkins, and Paul. On the number of Sudoku grids. Mathematics Today, April:94-95, 2014.

S. K. Jones, S. Perkins and P.A Roach, “The Structure of Reduced Sudoku Grids and the Sudoku Symmetry Group,” International Journal of Combinatorics, vol. 2012, Article ID 760310, 6 pages, 2012. doi:10.1155/2012/760310.

D. Williams, S.K. Jones, P. A. Roach, and S. Perkins, (2011) Blocking Intercalates In Sudoku Erasure Correcting Codes, International Journal of Computer Science (IJCS), Vol. 38, No. 3, pp. 183-191.

S. K. Jones, S. Perkins and P.A Roach,(2011) Properties, Isomorphisms and Enumeration of 2-Quasi-Magic Sudoku Grids, Discrete Mathematics, vol 311, pp. 1098-1110

Roach, P. A., Grimstead, I. J., Jones, S. K., and Perkins, S. (2009) A Knowledge-Rich Approach to the Rapid Enumeration of Quasi-Magic Sudoku Search Spaces. Proceedings of ICAART 2009, the 1st International Conference on Agents and Artificial Intelligence, Porto, Portugal, 19-21 January 2009, (INSTICC Press, Filipe, J., Fred, A. and Sharp, B. Eds.).246-254

S. K. Jones, P.A Roach, and S. Perkins (2007) Construction of Heuristics for a Search-Based Approach to Solving SuDoku. Accepted for Research and Development in Intelligent Systems XXIV: Proceedings of AI-2007, the Twenty-seventh SGAI International Conference on Artificial Intelligence, (Springer-Verlag, Bramer, M., Coenen, F., and Petridis, M. Eds),

On Cherlin’s Conjecture

This is an ongoing collaboration with Francesca Dalla Volta and Pablo Spiga (Milano—Bicocca); Martin Liebeck (Imperial); and Bianca Loda and Francis Hunt (USW).

In 2000, motivated by questions in model theory, Cherlin formulated a conjecture about finite BINARY permutation groups. Roughly speaking, a permutation group is binary if, by studying its action on pairs of points, one can deduce full information about the action. Cherlin’s conjecture asserts that all such permutation groups are known.

The conjecture has been reduced to a question about almost simple groups which means one can use the Classification of Finite Simple Groups. We have proved the conjecture for the alternating groups, the sporadic groups, and groups of Lie rank 1. We hope to finish what remains at some point soon!

On growth in groups

This is the subject of EPSRC grant EP/N010957/1.

On Growth in Groups. This is the subject of EPSRC grant EP/NO10957/1. The central question in this area is the PRODUCT DECOMPOSITION CONJECTURE: this conjecture asserts that given any finite simple group G and any subset S of G, you can write G as a product of conjugates of S “in the shortest possible way”. This conjecture has been proved in a number of special cases, including for groups of Lie type of bounded rank by myself, Ian Short, Laci Pyber and Endre Szabo. We would like to prove the full conjecture one day.

On Conway Groupoids

On Conway groupoids. In 1987, John Conway described a new way to construct the sporadic simple group $M_{12}$. His method involved a “game” played on $\mathbb{P3$, the finite projective plane of order $3$, and was remarkable because hitherto there was no known connection between the group $M{12}$ and the structure $\mathbb{P}_3$. In work with Neil Gillespie, Cheryl Praeger and Jason Semeraro, I have studied generalizations of this construction; this work has led to new constructions for a variety of families of groups.

Radio Frequency Assignment and Coding Theory

Some projects undertaken by Derek Smith on Radio Frequency Assignment and Coding Theory have been moved to the “Past Projects” section (or see “Documents”).